IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1203.6877.html
   My bibliography  Save this paper

The maximum maximum of a martingale with given $n$ marginals

Author

Listed:
  • Pierre Henry-Labord`ere
  • Jan Ob{l}'oj
  • Peter Spoida
  • Nizar Touzi

Abstract

We obtain bounds on the distribution of the maximum of a martingale with fixed marginals at finitely many intermediate times. The bounds are sharp and attained by a solution to $n$-marginal Skorokhod embedding problem in Ob{\l}\'oj and Spoida [An iterated Az\'ema-Yor type embedding for finitely many marginals (2013) Preprint]. It follows that their embedding maximizes the maximum among all other embeddings. Our motivating problem is superhedging lookback options under volatility uncertainty for an investor allowed to dynamically trade the underlying asset and statically trade European call options for all possible strikes and finitely-many maturities. We derive a pathwise inequality which induces the cheapest superhedging value, which extends the two-marginals pathwise inequality of Brown, Hobson and Rogers [Probab. Theory Related Fields 119 (2001) 558-578]. This inequality, proved by elementary arguments, is derived by following the stochastic control approach of Galichon, Henry-Labord\`ere and Touzi [Ann. Appl. Probab. 24 (2014) 312-336].

Suggested Citation

  • Pierre Henry-Labord`ere & Jan Ob{l}'oj & Peter Spoida & Nizar Touzi, 2012. "The maximum maximum of a martingale with given $n$ marginals," Papers 1203.6877, arXiv.org, revised Jan 2016.
    • Handle: RePEc:arx:papers:1203.6877
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1203.6877
    File Function: Latest version
    Download Restriction: no

    References listed on IDEAS

    as
    1. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
    2. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    3. Alexander Cox & Jan Obłój, 2011. "Robust pricing and hedging of double no-touch options," Finance and Stochastics, Springer, vol. 15(3), pages 573-605, September.
    4. Mathias Beiglbock & Pierre Henry-Labord`ere & Friedrich Penkner, 2011. "Model-independent Bounds for Option Prices: A Mass Transport Approach," Papers 1106.5929, arXiv.org, revised Feb 2013.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Matteo Burzoni & Marco Frittelli & Zhaoxu Hou & Marco Maggis & Jan Ob{l}'oj, 2016. "Pointwise Arbitrage Pricing Theory in Discrete Time," Papers 1612.07618, arXiv.org, revised Feb 2018.
    2. Johannes Muhle-Karbe & Marcel Nutz, 2016. "A Risk-Neutral Equilibrium Leading to Uncertain Volatility Pricing," Papers 1612.09152, arXiv.org, revised Jan 2018.
    3. repec:eee:spapps:v:127:y:2017:i:9:p:3005-3013 is not listed on IDEAS
    4. Julio Backhoff Veraguas & Mathias Beiglbock & Martin Huesmann & Sigrid Kallblad, 2017. "Martingale Benamou--Brenier: a probabilistic perspective," Papers 1708.04869, arXiv.org.
    5. Mathias Beiglbock & Marcel Nutz & Nizar Touzi, 2015. "Complete Duality for Martingale Optimal Transport on the Line," Papers 1507.00671, arXiv.org, revised Jun 2016.
    6. Sebastian Herrmann & Florian Stebegg, 2017. "Robust Pricing and Hedging around the Globe," Papers 1707.08545, arXiv.org.
    7. Mathias Beiglboeck & Pierre Henry-Labordere & Nizar Touzi, 2017. "Monotone Martingale Transport Plans and Skorohod Embedding," Papers 1701.06779, arXiv.org.
    8. Pierre Henry-Labordère & Nizar Touzi, 2016. "An explicit martingale version of the one-dimensional Brenier theorem," Finance and Stochastics, Springer, vol. 20(3), pages 635-668, July.
    9. Marcel Nutz & Florian Stebegg & Xiaowei Tan, 2017. "Multiperiod Martingale Transport," Papers 1703.10588, arXiv.org.
    10. Marcel Nutz, 2013. "Superreplication under Model Uncertainty in Discrete Time," Papers 1301.3227, arXiv.org, revised Feb 2014.
    11. Henry-Labordère, Pierre & Tan, Xiaolu & Touzi, Nizar, 2016. "An explicit martingale version of the one-dimensional Brenier’s Theorem with full marginals constraint," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2800-2834.
    12. Mathias Beiglboeck & Alexander Cox & Martin Huesmann, 2017. "The geometry of multi-marginal Skorokhod Embedding," Papers 1705.09505, arXiv.org.
    13. Marcel Nutz & Florian Stebegg, 2016. "Canonical Supermartingale Couplings," Papers 1609.02867, arXiv.org, revised Nov 2017.
    14. Stefan Gerhold & I. Cetin Gulum, 2016. "Consistency of option prices under bid-ask spreads," Papers 1608.05585, arXiv.org.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1203.6877. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (arXiv administrators). General contact details of provider: http://arxiv.org/ .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.